Gaussian Process Model With Variable Length Scales
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Solution Overview
Problem
Conventional Gaussian process models fail to adequately account for stronger variations in sub-regions of the input data space due to their assumption of constant length scales, leading to interpretation of local variations as measurement errors and resulting in reduced model precision, especially when dealing with high-dimensional training data.
Innovation Solution
The method involves ascertaining a point density function within the input data space to determine variable length scales for each input quantity, using the point density as a basis to generate a Gaussian process model with a covariance function that adapts to local variations, thereby improving model precision by increasing point density in areas with greater-than-average output quantity variations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional Gaussian process models with constant length scales are used, then the model structure remains simple, but the model precision deteriorates in sub-areas with stronger variations
Solution Approach 1:
The patent applies local quality by making the length scale parameter position-dependent rather than constant. The covariance function uses a length scale l(x) that varies with position in the input space, allowing the model to adapt to local variations in data density and output variations. This resolves the contradiction by improving precision in sub-areas with stronger variations while maintaining a relatively simple model structure based on the squared exponential covariance function.
2Measurement precision
If parameterized length scale functions are used to capture local variations, then the model precision improves, but the computational complexity increases considerably
Solution Approach 1:
The patent changes the length scale parameter from a constant to a position-dependent function l(x). This parameter change allows the model to capture local variations in the data without requiring complex parameterized functions. The simplified approach of using point density as the basis for length scale determination reduces computational complexity compared to conventional parameterized approaches while maintaining improved precision.
3Measurement precision
If parameterized length scale functions are used, then the model can capture local variations, but the number of training data points required increases
Solution Approach 1:
The patent changes the length scale parameter to be position-dependent, which allows the model to capture local variations more efficiently. This parameter change reduces the number of training data points required compared to parameterized length scale functions because the point density itself provides the necessary information about local variations, eliminating the need for additional data to estimate complex length scale parameters.
4Ease of manufacture
If conventional Gaussian process models are used, then the computational process remains simple, but local strong variations are interpreted as measurement errors and removed by smoothing
Solution Approach 1:
The patent applies local quality by making the length scale parameter position-dependent through the point density function. This allows the model to preserve local strong variations by adapting the smoothing scale to the local data density, rather than applying uniform smoothing that removes important local features. The computational process remains relatively simple while improving precision.
Data Source
AI summary
A method for creating a Gaussian process model as a data-based functional model for an output quantity that is to be modeled, based on training data in an input data space, including providing training data having training data points and output values, assigned to the training data points, of one or more output quantities; ascertaining a point density that is a function of the position of the training data points in the input data space; ascertaining a length scale function for each input quantity of the training data as a function of the point density; and generating a Gaussian process model from the training data and the output data of the output quantity to be modeled, based on the ascertained length scale functions.